Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T23:20:20.005Z Has data issue: false hasContentIssue false
  • English
  • Français

An Efficient Algorithm to Construct an Orthonormal Basis for the Extended Krylov Subspace

Published online by Cambridge University Press:  28 May 2015

Akira Imakura
Akira Imakura*
Affiliation:
Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1, Tennodai Tsukuba-city, Ibaraki 305-8573, Japan
*
Corresponding author. Email address: imakura@cs.tsukuba.ac.jp
Get access

Abstract

Subspace projection methods based on the Krylov subspace using powers of a matrix A have often been standard for solving large matrix computations in many areas of application. Recently, projection methods based on the extended Krylov subspace using powers of A and A−1 have attracted attention, particularly for functions of a matrix times a vector and matrix equations. In this article, we propose an efficient algorithm for constructing an orthonormal basis for the extended Krylov subspace. Numerical experiments indicate that this algorithm has less computational cost and approximately the same accuracy as the traditional algorithm.

Keywords

65F1065F2565F3015A06Extended Krylov subspaceKrylov subspacesubspace projection methodsorthonormal basislinear systems
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 3 , August 2014 , pp. 267 - 282
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Davis, T.A., The University of Florida Sparse Matrix Collection, http://www.cise.ufl.edu/research/sparse/matrices/.Google Scholar
[2]Druskin, V. and Knizhnerman, L., Extended Krylov subspaces: approximation of the matrix square root and related functions, SIAM J. Matrix Anal. Appl. 19, 755–771 (1998).CrossRefGoogle Scholar
[3]Golub, G.H. and van Loan, C.F., Matrix Computations, 3rd edition, The Johns Hopkins University Press, Baltimore (1997).Google Scholar
[4]Hestenes, M.R. and Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Nati. Bur. Stat. 49, 409–436 (1952).CrossRefGoogle Scholar
[5]Heyouni, M., Extended Arnoldi methods for large Sylvester matrix equations, Technical report, L.M.P.A. (2008).Google Scholar
[6]Heyouni, M. and Jbilou, K., An extended block method for large-scale continuous-time algebraic Riccati equations. Technical report, L.M.P.A., Universitie du Littoral (2007).Google Scholar
[7]Higham, N.J., Functions of Matrices: Theoryand Computation, SIAM, Philadelphia (2008).CrossRefGoogle Scholar
[8]Knizhnerman, L. and Simoncini, V., A new investigation of the extended Krylov subspace method for matrix function evaluations, Numer. Linear Alg. Appl. 17, 615–638 (2010).Google Scholar
[9]Knizhnerman, L. and Simoncini, V., Convergence analysis of the extended Krylov subspace method for the Lyapunov equation, Numer. Math. 118, 567–586 (2011).Google Scholar
[10]Meijerink, J.A. and van der Vorst, A., An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comut. 31, 148–162 (1977).Google Scholar
[11]Saad, Y., Krylov subspace methods for solving large unsymmetric linear systems, Math. Comput. 37, 105–126 (1981).Google Scholar
[12]Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia (2003).CrossRefGoogle Scholar
[13]Saad, Y., Numerical Methods for Large Eigenvalue Problems, 2nd edition, SIAM, Philadelphia (2011).Google Scholar
[14]Saad, Y. and Schultz, M.H., GMRES: A generalised minimal residual algorithm for solving non-symmetric linear systems, SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).Google Scholar
[15]Simoncini, V., A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput. 29, 1268–1288 (2007).Google Scholar
[16]Simoncini, V., Extended Krylov subspace for parameter dependent systems, Appl. Numer. Math. 60, 550–560 (2010).Google Scholar
[17]Sugihara, M. and Murota, K., Theoretical Numerical Linear Algebra, Iwanami, Tokyo (2009), in Japanese.Google Scholar
[18]van der Vorst, H.A., Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13, 631–644 (1992).Google Scholar